The completed Kirillov model and local-global compatibility for functions on Igusa varieties

TL;DR

This paper connects the cuspidal functions on Igusa varieties for GL(2) with the classical Kirillov model, establishing local-global compatibility and proposing conjectures for broader p-adic automorphic forms.

Contribution

It describes cuspidal functions on Igusa varieties as a completion of the Kirillov model and proves a weak local-global compatibility theorem for classical modular forms.

Findings

Identification of cuspidal functions with Kirillov model completion

Establishment of weak local-global compatibility for eigenspaces

Conjectures on Hida theory and compatibility for general Igusa varieties

Abstract

We describe the cuspidal functions $\mathbb{V}_b^{\mathrm{cusp}}$ on the ordinary Caraiani-Scholze Igusa variety for $\mathrm{GL}_2$ as a completion of the smooth Kirillov model for classical cuspidal modular forms, and identify a variant of Hida's ordinary $p$-adic modular forms with the coinvariants of an action of $\tilde{\mu}_{p^\infty}$ on $\mathbb{V}_b^{\mathrm{cusp}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in $\mathbb{V}_b^{\mathrm{cusp}}$ associated to classical cuspidal modular forms. Based on these results, we conjecture an analog of Hida theory and an associated local-global compatibility for functions on more general Caraiani-Scholze Igusa varieties, which are natural spaces of $p$-adic automorphic forms.